Let be a set. We define the power of in a language-theoretical manner as
for all , and
We define the sets and as
The elements of are called words on , and is called the empty word on .
We define the juxtaposition of two words as
where and , with for each and . It is easy to see that the juxtaposition is associative, so if we equip and with it we obtain respectively a semigroup and a monoid. Moreover, is the free semigroup on and is the free monoid on , in the sense that they solve the following universal mapping problem: given a semigroup (resp. a monoid ) and a map (resp. ), a semigroup homomorphism (resp. a monoid homomorphism ) exists such that the following diagram commutes:
- 1 J.M. Howie, Fundamentals of Semigroup Theory, Oxford University Press, Oxford, 1991.
|Date of creation||2013-03-22 16:11:41|
|Last modified on||2013-03-22 16:11:41|
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